AI 中文总结
本文研究贫乏不可约全纯辛流形，证明其自然锥重合且无全纯叶理，并利用Riemann-Roch多项式根给出检测MBM类及有理曲线存在的统一准则，同时刻画了此类流形的形变空间。
AI 中文摘要
我们研究贫乏不可约全纯辛流形，即不包含有理曲线也不包含余维一子簇的流形。我们证明，对于此类流形，\\(H^{1,1}(X,\mathbb R)\\) 中的几个自然锥重合，从而给出强刚性推论。特别地，贫乏椭圆不可约全纯辛流形不允许任何全纯叶理。利用这一刚性性质，我们给出一个准则，用于从任意不可约全纯辛流形上的Riemann-Roch多项式的根中检测单值双有理极小类。对每个本原负类，我们关联一个仅依赖于形变类型及其Beauville-Bogomolov-Fujiki平方的多项式。在正实根的简单根假设下，数\\(1\\)在这些根中的位置意味着当该类为\\((1,1)\\)型时，该类是MBM类。这给出了不可约全纯辛流形上存在有理曲线的统一根论充分条件。然后，我们用形变空间描述贫乏椭圆不可约全纯辛流形，并证明模空间的每个连通分量都包含一个具有正Picard秩的贫乏不可约全纯辛流形。
英文摘要
We study poor irreducible holomorphic symplectic manifolds, namely those containing no rational curves and no codimension-one subvarieties. We show that, for such manifolds, several natural cones in \(H^{1,1}(X,\mathbb R)\) coincide, giving strong rigidity consequences. In particular, poor elliptic irreducible holomorphic symplectic manifolds admit no holomorphic foliations. Using this rigidity property, we give a criterion for detecting monodromy birationally minimal classes from the roots of Riemann--Roch polynomials on any irreducible holomorphic symplectic manifold. To each primitive negative class we associate a polynomial depending only on the deformation type and on its Beauville--Bogomolov--Fujiki square. Under a simple-root assumption on the positive real roots, the position of the number \(1\) among these roots implies that the class is MBM whenever it is of type \((1,1)\). This gives a uniform root-theoretic sufficient condition for the existence of rational curves on irreducible holomorphic symplectic manifolds. We then describe poor elliptic irreducible holomorphic symplectic manifolds in terms of their deformation spaces and show that every connected component of the moduli space contains a poor irreducible holomorphic symplectic manifold of positive Picard rank.
